In a year N, the 320th day of the year is Thursday. In the year N+1, the 206th…

Day-of-week problems are solved with modular arithmetic: any two dates exactly 7 days apart fall on the same weekday, so the weekday shift between two dates equals (days elapsed) mod 7. A common year has 365 days (52 weeks + 1 day, so the same calendar date shifts by 1 weekday the next year), while a leap year has 366 days (a shift of 2 weekdays). Leap years are never consecutive.

1. Find the day-count gap from the 320th day of Year N to the 206th day of Year N+1: gap = (days in Year N minus 320) plus 206. If Year N is a common year (365 days), gap = 45 + 206 = 251, and 251 mod 7 = 6 -- a 6-day shift, so the two dates could not both be Thursday.

2. If Year N is a leap year (366 days), gap = 46 + 206 = 252, and 252 mod 7 = 0 -- a whole number of weeks, so both dates land on the same weekday. This matches the given data (both Thursday), so Year N must be a leap year.

3. Since Year N is a leap year, Year N-1 (the year immediately before it) is a common year, because leap years never occur in consecutive years -- so Year N-1 has 365 days.

4. Find the day-count gap from the 168th day of Year N-1 to the 320th day of Year N: gap = (365 minus 168) plus 320 = 197 + 320 = 517 days, i.e., 73 weeks + 6 days. So the 320th day of Year N falls exactly 6 weekdays after the 168th day of Year N-1. Since the 320th day of Year N is Thursday, the 168th day of Year N-1 is 6 weekdays earlier, i.e., Friday.

Cross-check: counting forward six weekdays from Friday -- Saturday, Sunday, Monday, Tuesday, Wednesday, Thursday -- returns to Thursday, confirming consistency with the given 320th-day-of-Year-N data.

Result: the 168th day of Year N-1 is Friday.